Analysis of HOD for Admissible Structures
Jan Kruschewski, Farmer Schlutzenberg
TL;DR
The paper develops an HOD-analysis in an admissible-set context under the presence of a Woodin cardinal by replacing the determinacy model with $\Sigma_n$-KP premice. It builds an $\mathcal{M}^{\text{ad}}$-like mouse, constructs a direct-limit framework $M_\infty$ via external and internal covering systems, and proves that a natural $H$ inside $L_\alpha[x,G]$ satisfies $H = M_\infty[\Sigma_0]$. Key results include that $\delta_\infty$ remains Woodin in $M_\infty[\Sigma_0]$, and that $M_\infty[*]$ is a ground for $L_\alpha[x]$, alongside a precise identification of $\Sigma_n$-HOD with $M_\infty[*]$ in the forcing extension. The work extends HOD-analysis from determinacy models to admissible structures, yielding fine-structural inner models and a robust framework for definability, iterability, and ground-model analysis in the presence of Woodin cardinals.
Abstract
Let $n \geq 1$ and assume that there is a Woodin cardinal. For $x \in \mathbb{R}$ let $α_x$ be the least $β$ such that \[ L_β[x] \models Σ_n \text{-KP} + \exists κ(``κ\text{ is inaccessible and }κ^+ \text{ exists}"). \] We adapt the analysis of $\text{HOD}^{L[x,G]}$ as a strategy mouse to $L_{α_x}[x,G]$ for a cone of reals $x$. That is, we identify a mouse $\mathcal{M}^{\text{n-ad}}$ and define a class $H \subseteq L_{α_x}[x,G]$ as a natural analogue of $\text{HOD}^{L[x,G]} \subseteq L[x,G]$, and show that $H = M_\infty[Σ_0]$, where $M_\infty$ is an iterate of $\mathcal{M}^{\text{n-ad}}$ and $Σ_0$ a fragment of its iteration strategy.